We prove a partial regularity result for local minimizers u : \mathbb{R}^{n}\supset\Omega\rightarrow\mathbb{R}^{M} of the variational integral J(u,\Omega)=\int_{\Omega}f(\nabla^{k}u)dx, where k is any integer and f is a strictly convex integrand of anisotropic (p,q)-growth with exponents satisfying the condition q < p(1 + 2/n). This is some extension of the regularity theorem obtained in [BF2] for the case n = 2.